SAVE THE GAME AFTER ACCEPTING THIS QUEST!!! You should only be able to complete this quest once, but it's a little buggy in this respect. Report to the correct house at the correct time to meet your client. As soon as you leave the house, your client will be killed and you will be given a letter to deliver to Queen Akorithi in Sentinel. You will also get to fight two mages and three or four archers.
There is no way to avoid having your client killed. I have even gone so far as to anchor a "Recall" right in front of Akorithi and then accept the quest and the client still dies as soon as I leave the house (and the Mages now use spells because I'm in a dungeon).
If you take the quest early enough in the day, you should get a letter telling you to meet someone at a local tavern. If you go to this meeting you will be given an amulet that will supposedly protect you from the Necromancers. In reality, if the amulet is in your possession at the time you leave the house, you will be teleported into a random dungeon. The message is a little buggy, so if you want to get it before meeting the client, activate "fast travel" and it will generate.
OK, your client is dead and you have the letter. You now have a decision to make. You can deliver the letter to Sentinel where you will be rewarded with some gold. You can deliver the letter to Lord K'avar at the palace mentioned in the letter and he will give you a reward. You can talk to Lord K'avar at the palace, but not give him the letter. This will cause him to disappear and reappear in the dungeon mentioned in the letter. Or you can ignore the whole thing.
In any event, you will be attacked by three archers once or twice per day until the quest is completed or expires. This can be a very good thing because each archer carries goodies, plus the same goodies as all of the other archers before him. By the time two or three weeks have passed, they are literally carrying a ton of goodies. Most of it is junk (leather, chain, iron weapons, etc.), but at high levels, they will be carrying a few really good items. If you are going to try to collect all of it, be aware that the game does not like to keep track of that much stuff and becomes very unstable. I have found that the best way to deal with it is to either sell it all as soon as possible or just pick out the good stuff and let the rest disappear when the body does.
Now for dealing with the quest options.
If you take the letter to Sentinel, Lord K'avar will be arrested, you will get your gold and be sent on your way. A few game weeks later, you will get a message from the queen telling you that K'avar has escaped and she wants you to go track him down and bring him back to justice (you should be able to capture him in this scenario). You get a second reward for doing this.
If you talk to K'avar but do not give him the letter, the queen will ask you to go track him down in the dungeon mentioned in the letter (Castle Necromoghan, if you accept the quest in Daggerfall province). You will not be able to capture him, but it is possible for him to escape if you don't kill him quickly. You will have about 30 days to do this, and you will get the archers every day until the time expires or K'avar is dead. If he escapes, you are supposed to get a message a few weeks later that he is in the dungeon at Wayrest, but this doesn't always work.
If you choose to give the letter to K'avar, you will get a reward. A few weeks later you will get a message to help him escape from Castle Sentinel before he is executed.
If you choose to ignore the whole thing, you will get to fight the archers for about a month, but they will quit eventually.
Successfully completing the quest will give you about a 20 point rep boost in Sentinel.
Now for the really fun part. What happens if you accept the quest after successfully completing it the first time? The answer is "a major headache". K'avar has already been killed, so he will not be at the Palace and will not appear in the dungeon. The only thing you can do in this instance is to immediately deliver the letter to the queen and weather the onslaught of archers. Interestingly enough, Lord K'avar is an NPC in Castle Sentinel. Even after completing the quest, he is still there (a really good trick for a dead guy) and you can still talk to him (he'll even offer you the occasional quest if you let your rep with him zero out again).